5: Calculus of Variations
- Page ID
- 9592
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The prior chapters have focussed on the intuitive Newtonian approach to classical mechanics, which is based on vector quantities like force, momentum, and acceleration. Newtonian mechanics leads to second-order differential equations of motion. The calculus of variations underlies a powerful alternative approach to classical mechanics that is based on identifying the path that minimizes an integral quantity. This integral variational approach was first championed by Gottfried Wilhelm Leibniz, contemporaneously with Newton’s development of the differential approach to classical mechanics.
- 5.1: Introduction to the Calculus of Variations
- During the 18th century, Bernoulli, who was a student of Leibniz, developed the field of variational calculus which underlies the integral variational approach to mechanics. He solved the brachistochrone problem which involves finding the path for which the transit time between two points is the shortest.
- 5.2: Euler’s Differential Equation
- The calculus of variations, presented here, underlies the powerful variational approaches that were developed for classical mechanics. Variational calculus, developed for classical mechanics, now has become an essential approach to many other disciplines in science, engineering, economics, and medicine.
- 5.3: Applications of Euler’s Equation
- The Brachistochrone problem involves finding the path having the minimum transit time between two points. The Brachistochrone problem stimulated the development of the calculus of variations by John Bernoulli and Euler.
- 5.4: Selection of the Independent Variable
- A wide selection of variables can be chosen as the independent variable for variational calculus. Selecting which variable to use as the independent variable does not change the physics of a problem, but some selections can simplify the mathematics for obtaining an analytic solution. The following example of a cylindrically-symmetric soap-bubble surface formed by blowing a soap bubble that stretches between two circular hoops, illustrates the importance of the independent variable.
- 5.5: Functions with Several Independent Variables
- The discussion has focussed on systems having only a single function y(x) such that the functional is an extremum. It is more common to have a functional that is dependent upon several independent variables f[y1(x),y′1(x),y2(x),y′2(x),....;x].
- 5.6: Euler’s Integral Equation
- An integral form of the Euler differential equation can be written which is useful for cases when the function f does not depend explicitly on the independent variable x.
- 5.7: Constrained Variational Systems
- Holonomic constraints couple coordinates for the system.
- 5.8: Generalized coordinates in Variational Calculus
- Generalized coordinates allow embedding constraint forces which simplifies the solution.
- 5.9: Lagrange multipliers for Holonomic Constraints
- The Lagrange multiplier technique provides a powerful, and elegant, way to handle holonomic constraints using Euler’s equations. The general method of Lagrange multipliers for n variables, with m constraints, is best introduced using Bernoulli’s ingenious exploitation of virtual infinitessimal displacements, which Lagrange signified by the symbol δ.
- 5.10: Geodesic
- The geodesic is defined as the shortest path between two fixed points for motion that is constrained to lie on a surface. Variational calculus provides a powerful approach for determining the equations of motion constrained to follow a geodesic.
- 5.11: Variational Approach to Classical Mechanics
- This chapter illustrates that variational principles provide a means of deriving more detailed information, such as the trajectories for the motion between given initial and final conditions, by requiring that scalar functionals have extrema values. For example, the solution of the brachistochrone problem determined the trajectory having the minimum transit time, based on only the magnitudes of the kinetic and gravitational potential energies.
Thumbnail: Minimizing function and trial functions. (CC BY-SA 2.5; Banerjee).